Well, it is of one of the forms $6m, 6m+1, 6m+2, 6m+3, 6m+4, 6m+5$, as these are the only possible choices. All but $6m+1, 6m+5$ are not prime for all $m \geq 1$, so we can eliminate all of those. If it is of the form $6m + 5$, then it is of the form $3n + 2$ (let $n = 2m$). Our assumption is that it is of the form $3n+1$, so that is not possible. Therefore, it is of the form $6m+1$ for some $m$.